Abstract
In this paper, a combination of algebraic and topological methods are applied to obtain new and structural results on harmonic rings. Especially, it is shown that if a Gelfand ring A modulo its Jacobson radical is a zero dimensional ring, then A is a clean ring. It is also proved that, for a given Gelfand ring A, then the retraction map \({\text {Spec}}(A)\rightarrow {\text {Max}}(A)\) is flat continuous if and only if A modulo its Jacobson radical is a zero dimensional ring. Dually, it is proved that for a given mp-ring A, then the retraction map \({\text {Spec}}(A)\rightarrow {\text {Min}}(A)\) is Zariski continuous if and only if \({\text {Min}}(A)\) is Zariski compact. New criteria for zero dimensional rings, mp-rings and Gelfand rings are given. The new notion of lessened ring is introduced and studied which generalizes “reduced ring” notion. Especially, a technical result is obtained which states that the product of a family of rings is a lessened ring if and only if each factor is a lessened ring. As another result in this spirit, the structure of locally lessened mp-rings is also characterized. Finally, it is characterized that a given ring A is a finite product of lessened quasi-prime rings if and only if \({\text {Ker}}\pi _{\mathfrak {p}}\) is a finitely generated and idempotent ideal for all \(\mathfrak {p}\in {\text {Min}}(A)\).
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Tarizadeh, A., Aghajani, M. Structural results on harmonic rings and lessened rings. Beitr Algebra Geom 62, 927–943 (2021). https://doi.org/10.1007/s13366-020-00556-x
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DOI: https://doi.org/10.1007/s13366-020-00556-x